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Study hours and grades The following table shows data on study hours per week and the effect on grades, with expected cell counts given underneath the observed counts for 200 college students in a study conducted by Washington's Public Interest Research Group (PIRG). \begin{tabular}{lcllc} \hline & \multicolumn{3}{c} { Effect on grades } & \\ \cline { 2 - 4 } Study hours per week & Positive & None & Negative & Total \\ \hline \(1-15\) & 26 & 50 & 14 & 90 \\ & 23.9 & 43.2 & 23.0 & \\ \(16-24\) & 16 & 27 & 17 & 60 \\ & 15.9 & 28.8 & 15.3 & \\ \(25-34\) & 11 & 19 & 20 & 50 \\ & 13.3 & 24.0 & 12.8 & \\ Total & 53 & 96 & 51 & 200 \\ \hline \end{tabular} 2002 ) (Source: USA Today, April 17 , a. Suppose the variables were independent. Explain what this means in this context. b. Explain what is meant by an expected cell count. Show how to get the expected cell count for the first cell, for which the observed count is 26 . c. Compare the expected cell frequencies to the observed counts. Based on this, what is the profile of subjects who tend to have (i) positive effect on grades than independence predicts and (ii) negative effect on grades than independence predicts.

Step by step solution

01

Understanding Independence

When two variables are independent, the distribution of one variable does not affect the distribution of the other. In this context, it means study hours per week have no effect on the grades.

02

Define Expected Cell Count

The expected cell count is the count we would expect if the variables were independent. It is calculated by \[ E = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}} \] This formula distributes the total proportionally to each cell assuming independence.

03

Calculate Expected Cell Count for First Cell

Using the formula, for the first cell (Study hours: 1-15, Positive effect): \[ E = \frac{90 \times 53}{200} = 23.85 \] Rounded we have approximately 23.9, which matches the given expected count.

04

Comparison of Observed and Expected Counts

Compare observed counts with expected counts: - For '1-15 hours', 26 observed (23.9 expected); more students than expected had a positive effect. - For '25-34 hours', 20 observed (12.8 expected); more students than expected had a negative effect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Cell Count

When analyzing data from an observational study, one important measure is the expected cell count. This represents what the counts in each category would be if the variables were independent. For instance, in a study analyzing study hours and their effect on grades, independence implies that study hours have no impact on grades.
To calculate an expected cell count, we use the formula: \[ E = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}} \] This formula helps distribute the total proportionally across each possible outcome, assuming there is no interaction between variables.
For example, to calculate the expected count for students studying 1-15 hours per week who have a positive effect on grades, we multiply the total number of students studying 1-15 hours (90) by the total number of students who had a positive grade effect (53), and divide by the overall total number of students (200): \[ E = \frac{90 \times 53}{200} = 23.85 \] Rounding gives us approximately 23.9, which is close to the given expected count and shows that more students than expected reported a positive effect.

Observational Study

An observational study is a type of research where the researcher observes subjects without manipulating the study environment. Instead of assigning treatments or interventions, researchers watch the natural occurrence of events. This is often used in social sciences and medicine to infer relationships between variables.
This type of study provides the data for analyzing relationships, such as study hours and their effect on academic grades. It gives us insights into existing patterns without stating cause and effect.
While observational studies can identify correlations and possible links between variables, they cannot, by themselves, prove causation. Many factors could influence the relationship between study hours and grades, including the quality of study or external support, like tutoring.
The benefit of these studies is in recognizing data trends. For instance, we can discern how different study durations correlate with perceived academic performance. However, it's crucial to remember that variables are not controlled, so inferring direct causation requires caution.

Data Analysis

Data analysis in the context of an observational study, like the one examining study hours' impact on grades, involves comparing observed data to expected data. This helps identify whether relationships exist beyond what would happen by chance.
In our study, comparing observed counts (e.g., 26 students with 1-15 study hours showing a positive effect) with expected counts (23.9 for the same group) can reveal statistical independence. If observed data consistently deviates from expected, it may suggest a potential relationship.
This comparison requires careful attention to:

• Differences in counts for each subgroup.
• Patterns that repeat across different group segments.
• Statistical measures, like chi-square tests, that quantify how likely observed differences are due to chance.

Data analysis helps us interpret the meaning behind numbers and decide whether patterns indicate actual relationships or random variance. For educators and policy makers, such analyses guide decision-making, highlighting areas needing further investigation or intervention.